Simultaneous control of deposition time and temperature of multi-zone furnaces

ABSTRACT

The present invention facilitates multi-zone furnace ( 102 ) based deposition processes by iteratively adjusting deposition time and zonal setpoint temperatures to mitigate deviations from desired target thickness(es). Coupled feedback loops are employed to update the deposition time ( 520 ) and the zonal setpoint temperatures ( 510 ) lot to lot and batch to batch while mitigating deviations fro the desired target thickness(es). Error checking is performed by computing an error metric ( 506 ) and only updating the setpoint temperatures on the error metric being within an acceptable value ( 508 ). Additionally, an excitation parameter ( 512 ) is determined that indicates variations in furnace operation.

FIELD OF THE INVENTION

The present invention relates generally to semiconductor devicefabrication, and more particularly, systems and methods for concurrentcontrol of deposition time and temperature in multi-zone furnaces.

BACKGROUND OF THE INVENTION

Semiconductor device fabrication involves using a number of fabricationprocesses to build a desired device. Generally, a semiconductor deviceis fabricated on a semiconductor material referred to as a substrate byforming layers or components, selectively patterning formed layers, andselectively implanting dopants into layers and/or the substrate in orderto achieve desired operational characteristics.

A typical process for forming a layer on a semiconductor device involvesplacing the device in a multi-zone furnace, supplying a depositionsource material (a gas), raising the furnace to a selected temperaturefor a selected period of time, and setting pressure at a suitable value,thereby causing deposition material to deposit on the device therebyforming the desired layer with a selected thickness.

Demand for semiconductor devices results in continuous demands forreduction in device and/or feature sizes as well as reductions inpermitted tolerances in formed layers. One known issue with regard tomulti-zone furnace deposition processes is that different zones orportions of the furnace can have different temperatures therebyresulting in varied deposition rates, which in turn lead to variationsin layer thicknesses. As permitted tolerances continue to shrink, suchvariations in thicknesses that occur in multi-zone furnace baseddeposition processes can become unacceptable. Some conventionalmechanisms have been employed to provide a tighter control of depositionrate. However, these conventional mechanisms are based on gas flow andthe like and can negatively impact the stoichiometry of the devicesbeing fabricated.

What is needed are systems and methods that improve uniformity in layerthicknesses for multi-zone furnace based depositions.

SUMMARY OF THE INVENTION

The following presents a simplified summary in order to provide a basicunderstanding of one or more aspects of the invention. This summary isnot an extensive overview of the invention, and is neither intended toidentify key or critical elements of the invention, nor to delineate thescope thereof. Rather, the primary purpose of the summary is to presentsome concepts of the invention in a simplified form as a prelude to themore detailed description that is presented later.

The present invention facilitates multi-zone furnace based depositionprocesses by iteratively adjusting deposition time and zonal setpointtemperatures to mitigate deviations from desired target thickness(es).Coupled feedback loops are employed to update the deposition time andthe zonal setpoint temperatures lot to lot and batch to batch whilemitigating deviations from the desired target thickness(es). Errorchecking is performed by computing an error metric and only updating thesetpoint temperatures on the error metric being within an acceptablevalue. Additionally, an excitation parameter is determined thatindicates variations in operation of the furnace, which is then employedto alter adjustments to the deposition time.

A system of the present invention includes a multi-zone furnace and acontroller. The furnace has multiple temperature zones and a centraltemperature zone, which is maintained at a nominal temperature duringdeposition. The furnace includes a number of slots into which wafers areplaced. Some of the wafers are monitor wafers that have probes attachedand provide thickness measurements during deposition processes. Thecontroller receives the thickness measurements and iteratively updatessetpoint temperatures for the multiple temperature zones as well asoverall deposition time. The location of the monitor wafers providingthe thickness measurements is accounted for. The setpoint temperaturesare updated using a developed thermal model and the received thicknessmeasurements. The deposition time is adjusted using the receivedthickness measurements and the updated setpoint temperatures.

To the accomplishment of the foregoing and related ends, the inventioncomprises the features hereinafter fully described and particularlypointed out in the claims. The following description and the annexeddrawings set forth in detail certain illustrative aspects andimplementations of the invention. These are indicative, however, of buta few of the various ways in which the principles of the invention maybe employed. Other objects, advantages and novel features of theinvention will become apparent from the following detailed descriptionof the invention when considered in conjunction with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a multi-zone furnace deposition system inaccordance with an aspect of the present invention.

FIG. 2 is a block diagram illustrating exemplary loading and location ofwafers in a multi-zone furnace in accordance with an aspect of thepresent invention.

FIG. 3 is a graph of a thermal model illustrating deposition rate versusslot position in accordance with an aspect of the present invention.

FIG. 4 is a diagram illustrating a furnace controller in accordance withan aspect of the present invention.

FIG. 5 is a flow diagram illustrating a method of controlling a furnacebased deposition process in accordance with an aspect of the presentinvention.

FIG. 6 is a graph illustrating deposition rate response across a furnaceload in accordance with an aspect of the present invention.

FIG. 7 is a graph illustrating deviation in deposition rate induced bytemperature setpoint variations in accordance with an aspect of thepresent invention.

FIG. 8 is a graph of a thermal model illustrating deposition rate versusslot position in accordance with an aspect of the present invention.

FIG. 9 is a graph illustrating an optimal corrected profile obtained inaccordance with an aspect of the present invention.

FIG. 10 is a graph illustrating suitable sample selection for monitoringtemperatures in a multi-zone furnace in accordance with an aspect of thepresent invention.

FIG. 11 is a graph illustrating a comparison between optimal andnon-optimal sampling in accordance with an aspect of the presentinvention.

FIG. 12 is a diagram illustrating a closed loop control scheme inaccordance with an aspect of the present invention.

FIG. 13 is a histogram illustrating distribution of probability ofsampling specific slots across the furnace per lot in accordance with anaspect of the present invention.

FIG. 14 is a histogram illustrating distribution of an error metricacross the furnace per lot in accordance with an aspect of the presentinvention.

FIG. 15 is a graph illustrating deviation from expected targetthicknesses for deposited isolation nitride in accordance with an aspectof the present invention.

FIG. 16 is a graph illustrating deviation from expected targets fordeposited sidewall nitride in accordance with an aspect of the presentinvention.

FIG. 17 is a graph illustrating deposition time adjustments for a numberof batches in accordance with an aspect of the present invention.

FIG. 18 is a graph illustrating temperature setpoint adjustments for anumber of batches in accordance with an aspect of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention will now be described with respect to theaccompanying drawings in which like numbered elements represent likeparts. The figures provided herewith and the accompanying description ofthe figures are merely provided for illustrative purposes. One ofordinary skill in the art should realize, based on the instantdescription, other implementations and methods for fabricating thedevices and structures illustrated in the figures and in the followingdescription.

The present invention facilitates multi-zone furnace based depositionprocesses by iteratively adjusting deposition time and zonal setpointtemperatures to mitigate deviations from desired target thickness(es).Coupled feedback loops are employed to update the deposition time andthe zonal setpoint temperatures batch to batch while mitigatingdeviations from the desired target thickness(es). Error checking isperformed by computing an error metric and only updating the setpointtemperatures on the error metric being within an acceptable value.Additionally, an excitation parameter is determined that indicatesproper or improper operation of the furnace, which is then employed toalter adjustments to the deposition time.

Turning now to FIG. 1, a block diagram of a multi-zone furnacedeposition system 100 in accordance with an aspect of the presentinvention is illustrated. The system is operative to perform thermalbased depositions of materials such as silicon-nitride via a lowpressure chemical vapor deposition from dichlorosilane and ammonia.Generally, the system 100 performs a deposition process at a depositionrate influenced by temperature for a specific amount of time to yield adeposited layer with a desired thickness. The system 100 is operable todeposit layers with varied deposition rates (e.g., sidewalls andisolation layers) and/or target thicknesses during a single batch or runof the system 100.

The system 100 includes a furnace 102 in which wafers are placed toundergo deposition, a controller 114 that controls operation of thefurnace 102, a first metrology tool 116, and a second metrology tool118. The furnace 102 has 5 different temperature zones in this aspect,however it is appreciated that other aspects of the invention can have adifferent number of zones. The zones include a first zone 104 (alsoreferred to as a top zone), a second zone 106, a central zone 108, athird zone 110, and a fourth zone 112 (also referred to as a bottomzone). The central zone 108 is kept at a fixed, nominal temperatureduring deposition and serves as a reference to decouple the impact ofadjustments to deposition time and adjustments to temperature setpointsof the other zones. The first zone 104, the second zone 106, the thirdzone 110, and the fourth zone 112 are controlled to be heated toindependent set point temperatures. The furnace 102 includes a number ofslots into which product wafers, monitor wafers, and dummy wafers can beinserted. Typically, the slots of the furnace 102 are filled duringdeposition processes in order to improve thermal uniformity of thefurnace. Generally, the wafers placed in the slots of the furnace arereferred to as a batch or run of wafers. Additionally, one or more lotsof wafers, defined as wafers having a similar layer deposited, can bepresent in a single batch.

As stated above, the furnace 102 is controllable with respect to zonetemperature setpoints and deposition time. Additionally, othercontrollable parameters include pressure (e.g., typically at about 0.25to 2.0 torr for a LPCVD deposition), gas flow, gas partial pressure, andwafer spacing.

FIG. 2 is a block diagram illustrating exemplary loading and location ofwafers in a multi-zone furnace in accordance with an aspect of thepresent invention. Here, first dummy wafers are placed in slots 1 to 9,first product wafers are placed in slots 10 through 58, additionalproduct wafers are placed in slots 59 through 107, and additional dummywafers are placed in slots 108 through 119. The first dummy wafers arealso generally located within a temperature zone T1. The first productwafers are generally located within a temperature zone T2 and anothertemperature zone TC. The second product wafers are located in thetemperature zone TC and another temperature zone T3. The second dummywafers are generally located within a temperature zone T4.

Returning to FIG. 1, the controller 114 controls operation of thefurnace 102, as stated above. The controller 114 is operative to supplyindependent set points to the first zone, 104, the second zone 106, thecentral zone 108, the third zone 110, and the fourth zone 112 and adjustdeposition time in response to temperature and thickness measurementsamples.

The first metrology tool 116 identifies processing parameters for anumber of batches including batch 1 lot 1 120 and batch 2 lot 2 122.These parameters include desired deposition material, deposition rates,desired deposition thickness, thermal budgets, and the like. The secondmetrology tool 118 also identifies processing parameters for a number ofbatches including batch 2 lot 1 124. These parameters include desireddeposition material, desired deposition thickness, thermal budgets, andthe like.

During operation, a batch of wafers, such as batch 1 lot 1 120 isprocessed by placing wafers of that batch in the furnace in a suitablemanner, such as shown in FIG. 2. An associated metrology tool, such asthe first metrology tool 116, provides processing parameters for thecurrent batch to the controller. The controller 114 develops a model ofthe furnace 102 that simulates thermal behavior through the differenttemperature zones of the furnace as well as etch rates as a function oftemperature. The model is developed based on thickness measurements andresulting deposition rates. The model indicates for a given temperaturezone changes in deposition rates according to changes in temperature.Alternately, the controller 114 can employ an existing, previouslydeveloped model and contemplates coupling between zones. Subsequently,the controller 114 determines initial set-points for the temperaturezones based on the model and the processing parameters and determinesand sets values for other controllable parameters including pressure(e.g., typically at about 0.25 to 2.00 torr for a LPCVD deposition), gasflow, gas partial pressure, gas type, and wafer spacing.

As a deposition process proceeds, the controller 114 receives depositionrate (or thickness) measurements and may receive temperaturemeasurement(s) from the furnace 102. Probes located on wafers can beemployed to obtain the thickness measurements. Generally, a number ofmonitor wafers placed in a number of slots are employed to obtain thesemeasurements. The placement of the monitor wafers can be selective inorder to facilitate obtaining desired deposition thicknesses.Additionally, variations in actual thicknesses of deposited layers onproduct wafers can vary. Such variations can be at least partiallyaccounted for by the controller 114. These measurements, also referredto as samples, are then employed by the controller 114 as feedback todetermine adjustments, if any, to the temperature zone setpoints anddeposition time according to the temperature samples, the developedmodel, and the processing parameters. The controller 114 also includes asafety feature wherein large adjustments and/or adjustments based onlarge erroneous measurements and/or sampling error are mitigated bysetting limits for temperature samples and time adjustments andtemperature setpoint adjustments.

FIG. 3 is a graph of a thermal model illustrating deposition rate versusslot position in accordance with an aspect of the present invention. Thegraph shows slot position on an x-axis and change in deposition rate(Angstroms per minute). The provided graph is exemplary in nature and isprovided to illustrate a suitable deposition-thermal response model ofthe present invention.

The graph illustrates how a change in temperature affects a depositionrate over slot positions in a typical multi-zone furnace. Line 301illustrates change in deposition rate for a degree change in temperaturein a first or top zone across the slot positions 0 to 120, line 302illustrates change in deposition rate for a degree change in temperaturein a second or top central zone across the slot positions 0 to 120, line303 illustrates change in deposition rate for a degree change intemperature in a third or bottom central zone across the slot positions0 to 120, and line 304 illustrates change in deposition rate for adegree change in temperature in a fourth or bottom zone across the slotpositions 0 to 120. These above lines illustrate coupling betweendifferent zones of the furnace. For example, the line 303 shows thatearlier slots experience a reduction in deposition rate for a positivedegree change in the third temperature zone. In updating temperature(s)and/or deposition times, a controller of the present invention accountsfor this coupling between zones.

FIG. 4 is a diagram illustrating a controller 400 in accordance with anaspect of the present invention. The controller 400 can be employed inthe system 100 of FIG. 1 or other similar multi-zone deposition systemsin order to update temperature set-points and deposition times based onreceived temperature samples.

The controller 400 includes a model 402, a Kalman filter 404, and anupdater 406. The updater 406 receives a number (N) of thicknessmeasurements from product wafers (not shown) located within a multi-zonefurnace (not shown). The thickness measurements are employed to buildthe model 402, which models deposition rate response per degreetemperature change for each zone across slots of the furnace. The number(N) of measurements needed is dependent on the number of parametersbeing controlled. For example, controlling four temperature zones anddeposition time requires at least five measurements. During operation,the updater 406 continues to receive the thickness measurements andemploys the model 402 with the thickness measurements to determine zonetemperature setpoint adjustments. Additionally, the updater 406 setsparameters and values of the Kalman filter 404 (described in detailinfra) and employs the Kalman filter 404, the model 402, the thicknessmeasurements, and the adjusted zone temperature setpoint adjustments toupdate values associated with the Kalman filter 404 and determine adeposition time adjustment. The updater 406 iteratively adjusts the zonetemperature setpoints and the deposition time adjustment during runs andfrom batch to batch. It is appreciated that alternate aspects of theinvention can employ other types of filters for updating and/oradjusting deposition time. Additionally, a suitable Kalman filter isdescribed in detail infra.

FIG. 5 is a flow diagram illustrating a method 500 of controlling afurnace based deposition process in accordance with an aspect of thepresent invention. The method iteratively update temperature zonesetpoints and deposition times for the furnace during a run of a batchand also from batch to batch.

The method 500 begins at block 502 where a thermal model is built for acurrent batch. The model is built from a number of product wafers andthickness measurements obtained for them in various slots located withinthe furnace. The slots selected for thickness measurements, includinglocation and number, can vary from batch to batch. A determination ismade at block 504 as to whether the batch is complete or not (i.e.,whether additional thickness measurements remain to be incorporated inthe model). If it is not, the method 500 continues to block 502 wherethe model continues being built. Once the batch is complete and themodel built, an error metric is determined at block 506. The errormetric represents expected error induced across the furnace load due tomeasuring a limited number of wafers instead of all wafers, as well as,modeling error present in the thermal model built in block 502. Theerror metric can vary according to the slots selected to obtainmeasurements there from. A suitable mechanism for determining the errormetric is to utilize the magnitude of the smallest non-zero singularvalue of the model. Generally, a high value of the error metricindicates less estimated or expected error and a low value indicatesmore estimated or expected error.

A determination is then made as to whether the error metric is greaterthan a threshold value referred to as a dead band at block 508. The deadband value is a threshold value and is selected according to allowableexpected error and model error. Additionally, the dead band value isselected to ensure that enough information is present to allow updatingof temperatures. The dead band value improves stability and to mitigatesthe impact of subsampling error (i.e., not sampling every slot) on thefurnace load. An exemplary value of the dead band value is 0.05, howeveremployed values are implementation dependent.

If the error metric is greater than the dead band value, temperatureset-points for one or more of the zones of the furnace are updated atblock 510. Typically, a central temperature zone is maintained at anominal temperature and other temperature zones are adjusted withrespect to their setpoints.

The method 500 continues at block 512 wherein an excitation value isdetermined. The excitation value represents whether enough informationis available in the measurements being received to correctly estimateall the parameters the model and is determined by tracking the changesin the processes running on the furnace—in a specific case for exampleby tracking changes (e.g., in deposition material thickness targets frombatch to batch) in the targets being run. A high value of the excitationvalue indicates that measurements have enough information to updatemultiple model parameters and a low value indicates that if an attemptis made to update multiple parameters, there is a high likelihood thatthe updates will be prone to error. If the excitation is sufficient, asdetermined at block 514, the method proceeds to block 516 where a filteris updated with K=K1. If the excitation is insufficient, the method 500proceeds to block 518 wherein the filter is updated with K=K2. The K1 isthe gain used to update multiple parameters simultaneously; whereas theK2 gives preference to a single parameter as low excitation implies thatthere is not enough information in the measurements to resolve multipleparameters. Subsequently, a deposition time is determined according tothe updated temperatures, measurements, and the Kalman filter and thefurnace is updated with the updated deposition time at block 520.

The method 500 can be repeated to provide for updated temperaturesand/or deposition times during a deposition process.

The following discussions describe and illustrate, including inmathematical terms, computations and operations performed by the presentinvention. Methods and systems of the present invention can be employedto perform deposition of silicon nitride from dichlorosilane and ammoniavia a low pressure chemical vapor deposition as well as other materials.Such deposited silicon nitride can be employed in a number of instancesfor semiconductor device fabrication. For example, the depositiedsilicon nitride can be utilized as a polish stop for shallow trenchisolation (such nitride is referred to as isolation nitride and as aspacer for transistor gates, referred to as sidewall nitride.Controlling thickness of deposited films across a furnace load (see FIG.2 for an example), including silicon nitride, facilitates consistentdevice performance. A single furnace load, which is wafers loaded into afurnace, can comprise multiple lots and processing conditions should berelatively uniform and within designed parameters to ensure low lot tolot variation. It is noted that batch to batch variations should bemitigated as well as variations within a single batch.

One mechanism for improving uniformity is to alter flow rates of sourcematerials (e.g., ammonia). However, this can impact a deposited layer orfilms stoichiometry, which in turn affects the performance of subsequentetch process.

The present invention builds a model by using empirical data based onthickness measurements. Typically, monitor and/or product wafers areemployed with a single tool to develop a suitable model.

The present invention contemplates a variety of furnaces. However, thefollowing discussion references an exemplary furnace that can beemployed for 300 mm wafers, can run up to four 24 wafer lots in a singlebatch, and has five temperature zones. Each load is comprised of dummywafers, product wafers, and monitor wafers. Dummy wafers are wafers usedto fill up empty slots in a load in order to present a consistentthermal mass and in FIG. 2 are presenting slots 1 through 9 and 109 to119. Monitor wafers are run and present in order to monitor particlesand are present in slots 10, 59, and 108, the last of which can also beused for control purposes. Product wafers are placed in remaining spots.The present invention mitigates deviations from target thicknesses onthe product wafers. A controller controls the deposition time for eachbatch, which can hold up to four 24 wafer lots and temperature setpointsfor four different zones, top, top center, bottom center, and bottomzones. The center zone is kept fixed at a nominal deposition temperatureand servers as a reference to decouple the impact of deposition timechanges, and temperature changes.

The controller performs time and/or temperature control as twoindependent subproblems. For controlling deposition time, a Kalmanfilter is employed to simultaneously update processing time for bothisolation and sidewall depositions, which are run at different targetdeposition rates. For temperature control, characterization ofcontrollability of the multi-zone furnace is analyzed and temperaturesampling along a number of slots is performed. The controller performsupdates iteratively on a lot to lot basis as well as a batch to batchbasis.

Generally, a Kalman filter is a set of mathematical equations thatprovides an efficient computational (recursive) means to estimate thestate of a process, in a way that minimizes the mean of the squarederror. The filter is powerful in several aspects: it supportsestimations of past, present, and even future states, and it can do soeven when the precise nature of the modeled system is unknown.

Controlling Deposition Time

The following discussion illustrates, in mathematical terms, onemechanism employed to control deposition time in order to drivedeposited layers toward a target thickness. The discussion contemplatesthat multiple lots can be employed in a single batch and that,therefore, different slots and/or wafers can be driven to differenttarget thicknesses in the same batch.

Let nεI₊ denote the number of furnace slots (excluding those slotsoccupied by dummies) of interest in the furnace load. Furthermore, anymathematical operation on a matrix or vector by a scalar is assumed toapply term by term.

For the following discussion, the following notations are employed:

-   k denotes the run number.-   |·| denotes the 1-norm.-   ∥·∥ denotes the (induced) 2-norm.-   μ(x) denotes the average of any vector x.-   S⊂I^(n){xεI^(n): x(i)=0 or 1,i=1,2, . . . ,n}.-   s_(k)εS: s_(k)(i)=1 if slot i is measured (sampled). Else    s_(k)(i)=0. i=1,2, . . . ,n. This is referred to as a sampling plan.-   {overscore (s)}_(k)    s_(k) such that s_(k)(i)=1,i=1,2, . . . ,n.-   y_(k) ^(s) ^(k) ε    ^(s) ^(k) ^(|)I: vector of thickness measurements corresponding to    s_(k).-   t_(k): deposition time.-   r_(k) ^(s) ^(k) =y_(k) ^(s) ^(k) /t_(k)−μ(y_(k) ^(s) ^(k) /t_(k))ε    ^(s) ^(k) ^(|) denotes the normalized deposition rate corresponding    to sampling plan s_(k).-   T_(k)ε    ⁴=[T1T2T3T4]′.-   I_((a))=1 if condition a is true, else equals 0.-   For any matrix M εR^(n×n) let ρ(M) denote its spectral radius. Also,    for any matrix MεR^(m×n),let σ_(i)(M), i≧1, denote its i^(th)    largest singular value.

The aim of controlling deposition time is to drive the mean thickness ofa current batch to a thickness target (Γ_(k)). Since the process couldpotentially be running multiple targets, it is useful to express therelationship between the mean thickness and deposition time as:$\begin{matrix}{{\mu\left( y_{k}^{s_{k}} \right)} = {{{\theta_{k}(1)} + {{\theta_{k}(2)}t_{k}} + w_{k}}\quad = {{{\left\lbrack {1\quad t_{k}} \right\rbrack\begin{bmatrix}{\theta_{k}(1)} \\{\theta_{k}(2)}\end{bmatrix}} + w_{k}}\quad = {{u_{k}\theta_{k}} + w_{k}}}}} & (1)\end{matrix}$where w_(k)ε

is assumed to be zero mean i.i.d. distributed with cov(w)=R. Eq. (1) isa linear stochastic difference equation that estimates the meanthickness. θ_(k)(1) is referred to as a pattern offset and θ_(k)(2) is adeposition rate. The term w_(k) represents process, assumed to be randomwhite noise. Furthermore, it is assumed that:θ_(k+1)=θ^(k)+ν_(k) ,E(θ₀)={overscore (θ)}where ν_(k)εR² is zero mean i.i.d. (independent, identically distributedrandom variables) with cov(ν)=Q, and is independent of w_(k); and E(·)denotes the expectation operator. The term ν_(k) is measurement noise,θ_(k) is a vector of furnace parameters (e.g., thickness measurements,deposition rate, and the like) from (1) for a current batch and θ_(k+1), represents a vector of furnace parameters for a next batch. LetΣ₀=var(θ₀)+Q. As a result, the Kalman filter recursions can be expressedas:Σ_(k+1)=Σ_(k)−Σ_(k) u _(k)(u _(k)Σ_(k) u _(k) +R)⁻¹ u _(k)Σ_(k) +QK _(k)=Σ_(k) u _(k)(u _(k)Σ_(k) u _(k) +R)⁻¹{circumflex over (θ)}_(k+1)={circumflex over (θ)}_(k) +K _(k)(μ[y _(k)^(s) ^(k) ]−u _(k){circumflex over (θ)}_(k)),{circumflex over(θ)}₀={overscore (θ)}  (2)where {circumflex over (θ)}_(k) denotes the estimate of the actualparameter {overscore (θ)}_(k). The above equation (2) yields the minimumvariance estimate of θ_(k), i.e., one that minimizesE[(θ_(k)−{circumflex over (θ)}_(k))(θ_(k)−{circumflex over (θ)}_(k))′].Then at the start of any run k, given the target for the run (Γ_(k)),one can compute the deposition time by inverting the model (1) asfollows: $\begin{matrix}{t_{k} = {\frac{\Gamma_{k} - {{\hat{\theta}}_{k}(1)}}{{\hat{\theta}}_{k}(2)}.}} & (3)\end{matrix}$It is noted that there exists a tendency in fabrication to dedicate atool for running material at a specific process step (e.g., at isolationnitride deposition). This is done in order to simplify inventorymanagement and scheduling. The control mechanism of the presentinvention monitors for sufficient excitation of the Kalman filter, whichis a measure of whether enough information is available from themeasurements to reliably compute both parameters in {circumflex over(θ)}_(k). This can be accomplished by tracking the batch-to-batchvariation in the thickness target(s) (Γ_(k)). The Kalman filter can beturned off if sufficient variation in the process mix as indicated bychanges in the targets of the batches being run is not detected, andrevert to updating the model via integral control. In order to do thisconsider $\begin{matrix}{{{\xi_{k + 1} = {{\left( {1 - ɛ_{\xi}} \right)\xi_{k}} + {ɛ_{\xi}{\delta\left( {{\Gamma_{k} - \Gamma_{k - 1}},{\Delta\Gamma}} \right)}}}},{\xi_{0} = 1}}{where}{{\delta\left( {a,b} \right)}\overset{\bigwedge}{=}\left\{ \begin{matrix}{{{1\quad{if}\quad{a}} \geq b},} \\{0\quad{{else}.}}\end{matrix} \right.}} & (4)\end{matrix}$Thus, if batch to batch (or lot to lot) variation in thickness target|Γ_(k)−Γ_(k−1)| is less than a threshold amount ΔΓ the termδ(Γ_(k)−Γ_(k−1), ΔΓ) is equal to zero, which effectively “switches off”the Kalman filter within a limited number of batches.

-   Then, given τ_(ξ), such that 0<τ_(ξ)<1, and ε_(θ), with 0<ε_(θ)<2,    the gain K_(k) in Eq. (2), above, can be modified as follows:    $K_{k} = \left\{ \begin{matrix}    {{{\underset{k}{\Sigma}{u_{k}\left( {{u_{k}\underset{k}{\Sigma}u_{k}} + R} \right)}^{- 1}\quad{if}\quad\xi_{k}} \geq \tau_{\xi}},} \\    {\begin{bmatrix}    0 \\    {{ɛ_{\theta}/t_{k}}\quad}    \end{bmatrix}\quad{{else}.}}    \end{matrix} \right.$    τ_(ξ) and ε_(θ) represent threshold values for switching between the    two gain values in the above equation and a filter factor for    tracking the excitation, respectively. Combining (2) and (4) obtains    the following recursion for tracking the parameters in (1) for    determining deposition time. $\begin{matrix}    {{\underset{k + 1}{\Sigma} = {\underset{k}{\Sigma} - {I_{({\xi_{k} \geq \tau_{\xi}})} \cdot \left( {{\underset{k}{\Sigma}{u_{k}\left( {{u_{k}\underset{k}{\Sigma}u_{k}} + R} \right)}^{- 1}u_{k}\underset{k}{\Sigma}} - Q} \right)}}}{K_{k} = {{{I \cdot \underset{k}{\Sigma}}{u_{k}\left( {{u_{k}\underset{k}{\Sigma}u_{k}} + R} \right)}^{- 1}} + {I_{({\xi_{k} \geq \tau_{\xi}})} \cdot \begin{bmatrix}    {\quad 0} \\    {{ɛ_{\theta}/t_{k}}\quad}    \end{bmatrix}}}}{\xi_{k + 1} = {{\left( {1 - ɛ_{\xi}} \right)\xi_{k}} + {ɛ_{\xi}{\delta\left( {{\Gamma_{k} - \Gamma_{k - 1}},{\Delta\Gamma}} \right)}}}}{{\hat{\theta}}_{k + 1} = {{\hat{\theta}}_{k} + {K_{k}\left( {{\mu\left( y_{k}^{s_{k}} \right)} - {u_{k}{\hat{\theta}}_{k}}} \right)}}}} & (5)    \end{matrix}$

Eq. (5) can then be employed by a controller of the present invention todetermine deposition time for each run and to adjust the deposition timeduring runs of the batch by measuring deposition rates at one or moreproduct wafers within a batch and using the measured values for θ_(k).

Modification to the above recursion to account for coupling betweentemperature and rate is discussed below.

Control of Cross-Load Variation

Cross-load variation is a variation in temperature across wafers presentin a furnace for a single batch. As stated previously, wafers of a batchare loaded into a number of slots of a furnace, one wafer per slot.These cross-load variations can impact and alter deposition rates and,therefore, resulting thicknesses of deposited layers. Cross-loadvariation is controlled or mitigated by controlling temperaturesetpoints (T) of zones present within a furnace. This problem is posedas minimizing at any given run k the following:e _(k) ^({overscore (s)}) =∥r _(k) ^({overscore (s)})−μ(r _(k)^({overscore (s)}))∥where the entire furnace load is assumed to be sampled (i.e.,s={overscore (s)}) and the term r_(k) ^({overscore (s)}) representsmeasured temperatures. As a result, the above can be rewritten asminimizing:e _(k) ^({overscore (s)})=∥(I _(|{overscore (s)}|)−1_(|{overscore (s)}|)/|{overscore (s)}|)·r _(k) ^({overscore (s)})∥where the multiplierI_(|{overscore (s)}|)−1_(|{overscore (s)}|)/|{overscore (s)}| issingular. As a result, instead of solving the above equation, terms canbe removed leaving the simplified equation below.e _(k) ^({overscore (s)}) =∥r _(k) ^({overscore (s)})∥ with r _(k−1)^({overscore (s)}) assumed to be 0 mean.Let e_(k,opt) ^({overscore (s)}) denote the minimum value of this error.

Generally, it is not practical to measure every wafer in the furnaceload. Typically, only a subset of wafers in a furnace are measured(i.e., |s|<|{overscore (s)}|). Using such a subset results in anon-optimal rate profile across the furnace load referred to assubsampling error. Additionally, in this example, there are fourtemperatures to be determined and these temperatures can be solved fromfour temperature and/or thickness measurements, referred to as asampling plan. A suitable sampling plan (s*) should satisfys*εarg min_({sεS:|s|=4}) ∥e ^(s) −e _(k,opt) ^({overscore (s)})∥  (6)as well as to characterize this error. However, in practice, non-optimalsampling plans can be employed that do not meet the restrictions of Eq.(6), above. Such non-optimality is induced not only by productionsampling constraints, but also due to the need to simultaneously controldeposition time, in addition, to the four temperatures, which requiresat least five measurements to resolve the five unknowns. In this case,it is of interest to also characterize the penalty in terms of the errorin comparison to the optimal error obtained by considering all the slots(i.e., via {overscore (s)}). In order to address these issues, behaviorof the furnace to changes in temperature (T) can be investigated.

Temperature Response

The following discussion is an example of characterizing a temperatureresponse of a furnace and is provided for illustrative purposes. Forthis example, a sequence of furnace runs is carried out with monitorwafers distributed across the furnace load, and with dummy wafersoccupying the remaining slots in order to characterize the relationshipbetween the deposition rate and temperature. Product wafers are notemployed due to their high cost. Furthermore, all the runs are on asingle tool only. Given the time involved in a single run (typicallyabout 6 hours), only one side (positive) of the nominal temperatureprofile is explored. Table 1 shows the conditions considered. TABLE 1Sequence of one-sided temperature tests. Case T₁ T₂ T_(C) T₃ T₄ 1 0 0 00 0 2 + 0 0 0 0 3 + + 0 0 0 4 0 + 0 0 0 5 + 0 0 0 + 6 0 + 0 + 0 7 0 00 + + 8 0 0 0 + 0 9 0 0 0 0 +

For this example, there are five temperature zones T₁, T_(2, T) _(C),T_(3, and T) ₄, from one end of a furnace to another. In table 1, a “0”entry denotes that the zone was set to the nominal process temperatureT_(C), and a “+” entry denotes that the zone was set to a temperatureequal to T_(C)+5° C. It is assumed that the response to temperature islocally linear, and a negative (T_(C)−5° C. ) change would induce anopposite effect to that observed in the cases considered in Table 1.Generally, large changes in temperature are not expected underrun-to-run control, hence the response can be assumed to be locallylinear. Furthermore, any local nonlinearity can be treated as modelingerror whose effect can be made to vanish under the effect of feedback.Stability issues pertaining to such approximation are discussed infra.

FIG. 6 is a graph illustrating a measured deposition rate responseacross a furnace load in accordance with an aspect of the presentinvention. The graph is provided as an example to further illustrate thepresent invention. The deposition rate measured is as a deviation from amean deposition rate across the furnace load (represented by slotpositions) for the case where all temperature zones are fixed at thenominal setpoint value (T_(C)). The deposition rate is illustrated bythe y-axis and a slot position, from 0 to 120, is represented by thex-axis. Circles 601 show measured thickness along with a linear fit.Accordingly, FIG. 6 depicts a linear rate increase in deposition ratedeviation across the furnace load in this example.

FIG. 7 is a graph illustrating deviation in deposition rate induced bytemperature setpoint variation for cases 2, 3, and 4 in Table 1 inaccordance with an aspect of the present invention. The deposition rateis illustrated by a y-axis and a slot position, from 0 to 120, isrepresented by an x-axis. Here the deviation in deposition rate for eachcase is calculated as a difference in the observed rate profile ascompared to the nominal case, i.e., case 1.

Rate profile as deviation from nominal profile for setpoint changes isshown. Line 701 (+) illustrates a change in deposition rate response forcase 2, line 702 (□) illustrates a change in deposition rate responsefor case 3, line 703 (ο) illustrates a change in deposition rateresponse for case 4, and line 704 (*) illustrates a change in depositionrate for a sum of cases 2 and 4. The responses have been scaled tocorrespond to that for a 1° C. change.

As observed from the figure, the response to case 3 line 703, whichcombines the setpoint changes from cases 2 and 4, appears to behave asthe sum of the individual responses from cases 2 and 4 at the line 704.This is seen to hold well for other zonal combinations as well.

It is noted that error induced by assuming an additive zonal responsecan be ignored if temperature changes are expected to be relativelysmall. It is assumed that less than a degree Celsius of change occursbecause the equations are employed iteratively by the controller. Theimpact of this assumption on stability will be discussed further below.

FIG. 8 is a graph of a thermal model illustrating deposition rate versusslot position in accordance with an aspect of the present invention. Thegraph shows slot position on an x-axis and change in deposition rate(Angstroms per minute). The provided graph is exemplary in nature and isprovided to illustrate a suitable deposition-thermal response model ofthe present invention.

The graph illustrates how a change in temperature affects a depositionrate over slot positions in a typical multi-zone furnace. Line 801illustrates change in deposition rate for a degree change in temperaturein a first or top zone across the slot positions 0 to 120, line 802illustrates change in deposition rate for a degree change in temperaturein a second or top central zone across the slot positions 0 to 120, line803 illustrates change in deposition rate for a degree change intemperature in a third or bottom central zone across the slot positions0 to 120, and line 804 illustrates change in deposition rate for adegree change in temperature in a fourth or bottom zone across the slotpositions 0 to 120. These above lines illustrate coupling betweendifferent zones of the furnace. For example, the line 803 shows thatearlier slots experience a reduction in deposition rate for a positivedegree change in the third temperature zone. In updating temperature(s)and/or deposition times, a controller of the present invention accountsfor this coupling between zones.

The individual zonal responses are shown in FIG. 8, whereƒ^(i)εR^(|s|),i=1,2,3,4 denotes the rate profile across the furnace loadfor a degree change in the setpoint for zone i.

Under the additive zonal response assumption, the impact of temperaturechange (ΔT) can be expressed as a linear model:Δr ^({overscore (s)}) =F ^({overscore (s)}) ΔT  (7)where Δr^({overscore (s)})=r_(new) ^({overscore (s)})−r_(old)^({overscore (s)}), ΔT=T_(new)−T_(old), and F^({overscore (s)})ε

^(×4) is defined as followsF ^({overscore (s)})=[ƒ¹ƒ²ƒ³ƒ⁴]where ƒ^(i) (x) is the impact of zone i on slots x as shown in FIG. 8and F^({overscore (s)}) is a vector of zonal responses for a furnace.Significant interaction can be seen amongst the zones. For example,considering ƒ³, an increase in T₃ increases the rate in the lower halfof the furnace, but it also causes the rate in the upper half of thefurnace to drop. This should be considered in control operation in orderto avoid an oscillating control response, as indicated in FIG. 8.

In practice, each of the response curves in FIG. 8 can be modeled via apiece-wise cubic spline fit. For example,f¹(x) = (14.8 − 1.14(x − 8) + 0.0081(x − 8)³) ⋅ I_((x < 22)) + (1.12 − 0.66(x − 22) + 0.034(x − 22)² − 0.00052(x − 22)³) ⋅ I_((22 ≤ x < 36)) − (2.89 + 0.015(x − 36) − 0.012(x − 36)² + 0.00035(x − 36)³) ⋅ I_((36 ≤ x < 50)) − (1.54 − 0.15(x − 50) + 0.00046(x − 50)² + 0.000066(x − 50)³) ⋅ I_((50 ≤ x < 64)) + (0.27 + 0.097(x − 64) − 0.0032(x − 64)² − 0.0000043(x − 64)³) ⋅ I_((64 ≤ x < 78)) + (0.99 + 0.0042(x − 78) − 0.0034(x − 78)² + 0.00014(x − 78)³) ⋅ I_((78 ≤ x < 92)) + (0.75 − 0.011(x − 92) + 0.0023(x − 92)² − 0.000042(x − 92)³) ⋅ I_((92 ≤ x)).ƒ², ƒ³, and ƒ⁴ are defined similarly.

It is noted that for any general slot sample s, with |s|<|{overscore(s)}|, let F^(s)ε

^(s|×4) denote a matrix constructed by considering only those rows ofF^({overscore (s)}), i.e. F^({overscore (s)})(i,:) that correspond tos(i) =1.

Characterization of Furnace Controllability

The following discussion illustrates a characterization of the inherentcontrollability of a cross-furnace rate profile via zonal temperature asappreciated by the inventor of the present invention. The followingdiscussion focuses on control of temperature. Also controllingdeposition time induces non-optimality in temperature control,especially when the furnace load is sampled, i.e. not all furnace slotsof interest are measured. Given only four temperatures to manipulate,one will not, in general, be able to match the deposition rates acrossall wafers. This discussion attempts to capture this information interms of an attenuation gain, which can be potentially used to comparethe response of different furnaces.

Motivated by FIG. 6, assume that the initial rate response with allzones set to nominal temperature (T_(C)) is zero mean linear, i.e.${r_{0}^{\overset{\_}{s}} \in {\mathbb{R}}^{\overset{\_}{s}}},{{s.t.\quad{r_{0}^{\overset{\_}{s}}(i)}} = {\alpha\left( {i - i_{0}} \right)}},{{{with}\quad{\sum\limits_{i = 1}^{\overset{\_}{s}}{r_{0}^{\overset{\_}{s}}(i)}}} = 0},{\alpha > 0.}$For future use, define for any sεS,F ^(s†)ε

^(4×|s|) , s.t. F ^(s†) =[F ^(s′) F ^(s)]⁻¹ F ^(s′),where it is assumed that for any sεS considered, F^(s) has full columnrank and represents a zonal response of a furnace for less than all ofthe slots. This assumption will always be satisfied for any sεS forwhich temperature updates will be made via stability constraintsdescribed infra. As a result, the optimal temperature change (ΔT*) thatminimizes∥r ₀ ^({overscore (s)}) +F ^({overscore (s)}) ΔTοis given byΔT*=−F ^({overscore (s)}†) r ₀ ^({overscore (s)}).  (8)Given this, the new rate profile {circumflex over (r)}^({overscore (s)})is (via Eq. (7))${\hat{r}}^{\overset{\_}{s}} = {{r_{0}^{\overset{\_}{s}} + {F^{\overset{\_}{s}}\Delta\quad T^{*}}} = {\left( {I_{\overset{\_}{s}} - {F^{\overset{\_}{s}}F^{\overset{\_}{s}\dagger}}} \right){r_{0}^{\overset{\_}{s}}.}}}$Let L^(|{overscore (s)}|) denote the space of zero mean linear rateprofiles. Then define γ_(opt)>0 as $\begin{matrix}{{\gamma_{opt}\sup\limits_{r_{0}^{\overset{\_}{s}} \in L^{\overset{\_}{s}}}\frac{{{\hat{r}}^{\overset{\_}{s}}}}{{r_{0}^{\overset{\_}{s}}}}} = \frac{{\left( {I_{\overset{\_}{s}} - {F^{\overset{\_}{s}}F^{\overset{\_}{s}\dagger}}} \right)r_{0}^{\overset{\_}{s}}}}{r_{0}^{\overset{\_}{s}}}} & (9)\end{matrix}$for any r₀ ^({overscore (s)})εL^(|{overscore (s)}|), s.t. r₀^({overscore (s)})≠0. γ_(opt) is thus the optimal attenuation gain forany initial zero-mean linear rate profile. Furthermore, let {tilde over(r)}^({overscore (s)}) be any zero-mean nonlinear rate profile. Then onehas${r_{0}^{\overset{\_}{s}}} = {{{\left( {I_{\overset{\_}{s}} - {F^{\overset{\_}{s}}F^{\overset{\_}{s}\dagger}}} \right){\overset{\sim}{r}}^{\overset{\_}{s}}}}\quad = {{{{\left( {I_{\overset{\_}{s}} - {F^{\overset{\_}{s}}F^{\overset{\_}{s}\dagger}}} \right)\left( {{\overset{\sim}{r}}^{\overset{\_}{s}} - r_{0}^{\overset{\_}{s}} + r_{0}^{\overset{\_}{s}}} \right)}}\quad{for}\quad{any}\quad r_{0}^{\overset{\_}{s}}} \in {{L_{0}^{\overset{\_}{s}}\backslash\left\{ 0 \right\}} \leq {~~~~~~~~~~~~~~~~}{{\left( {I_{\overset{\_}{s}} - {F^{\overset{\_}{s}}F^{\overset{\_}{s}\dagger}}} \right)r_{0}^{\overset{\_}{s}}{{{{+ {{I_{\overset{\_}{s}} - {F^{\overset{\_}{s}}F^{\overset{\_}{s}\dagger}}}}}{{{\overset{\sim}{r}}^{\overset{\_}{s}} - r_{0}^{\overset{\_}{s}}}}} \leq {~~~~~~~~~~~~~~~}{{\gamma_{opt}{r_{0}^{\overset{\_}{s}}}} + {{{{\overset{\sim}{r}}^{\overset{\_}{s}} - r_{0}^{\overset{\_}{s}}}}.}}}}}}}}}$Pick r_(l) ^({overscore (s)}) as the best linear approximation of {tildeover (r)}^({overscore (s)}). Note that since {tilde over(r)}^({overscore (s)}) is zero-mean, it implies that r_(l)^({overscore (s)})εL^(|{overscore (s)}|). Then one obtains:$\frac{{\hat{r}}^{\overset{\_}{s}}}{{\overset{\sim}{r}}^{\overset{\_}{s}}} \leq {{\gamma_{opt}\frac{r_{l}^{\overset{\_}{s}}}{{\overset{\sim}{r}}^{\overset{\_}{s}}}} + \sqrt{1 - R^{2}}}$where R² is the fit error in approximating {tilde over(r)}^({overscore (s)}) by r_(l) ^({overscore (s)}).

FIG. 9 is a graph illustrating an optimal corrected profile obtainedfrom the furnace characteristics shown in FIG. 6 and FIG. 8. inaccordance with an aspect of the present invention. An x-axis representsslot position from 0 to 120 and a y-axis represents corrected depositionrate profile. Profile 901 yields an attenuation gain γ_(opt)=0.1863. Itis noted that, in general, this will not be a flat profile, and γ_(opt)can be used to compare the amount of control afforded by differentfurnaces.

Impact of Sampling

In the above discussion, it has been assumed that all positions in thefurnace are measured, and these obtained measurements are available foradjusting the zonal temperatures. In practice, it is generally feasibleto measure a limited number of wafers (e.g., one wafer per lot). Thispractice of sampling a limited number of wafers induces errors inestimating the temperature setpoint changes, and causes the rate profileacross the furnace load to deviate from the optimal profile (such as theprofile 901 shown in FIG. 9). The following discussion characterizes theerrors induced by sub-sampling the furnace load as appreciated by theinventor of the present invention. Once again, a zero-mean linearprofile is used and the resultant attenuation gain characterized interms of the optimal attenuation gain γ_(opt).

Let s, with |s|<|{overscore (s)}| denote the sampling plan. Let r₀^({overscore (s)}) denote the initial zero-mean linear profile, and letr₀ ^(s) denote this profile sub-sampled according to s. Let {circumflexover (r)}^({overscore (s)}) denote the optimal corrected profile. As aresult, the following equation is obtained:{circumflex over (r)} ^(s) =r ₀ ^(s) +F ^(s) ΔT*  (10)where ΔT* is the temperature profile from Eq. (8). Let r^(s) denote theoptimal corrected profile obtained by considering r₀ ^(s), and letΔT^(s) be the temperature profile, as defined by this equation:${\Delta\quad T^{s}} \in {\arg\quad{\min\limits_{{\Delta\quad T} \in R^{4}}{{{r_{0}^{s} + {F^{s}\Delta\quad T}}}.}}}$Thenr ^(s) =r ₀ ^(s) +F ^(s) ΔT ^(s).  (11)From the above, the following is obtained{circumflex over (r)} ^(s) −r ^(s) =F ^(s) [ΔT*−ΔT ^(s)].  (12)As a result, the below equation is obtained $\begin{matrix}{r^{\overset{\_}{s}} = {r_{0}^{\overset{\_}{s}} + {F^{\overset{\_}{s}}\Delta\quad T^{s}}}} \\{= {{\hat{r}}^{\overset{\_}{s}} + {F^{\overset{\_}{s}}\left( {{\Delta\quad T^{s}} - {\Delta\quad T^{*}}} \right)}}} \\{= {{\hat{r}}^{\overset{\_}{s}} + {F^{\overset{\_}{s}}{F^{s\quad\dagger}\left( {{\hat{r}}^{s} - r^{s}} \right)}}}} \\{= {{\hat{r}}^{\overset{\_}{s}} + {F^{\overset{\_}{s}}F^{s\quad\dagger}{{\hat{r}}^{s}.}}}}\end{matrix}$The last result following from the fact that r^(s) is in the null spaceof F^(s′). Which implies that the attenuation gain (γ_(s)) for thesampled case is bounded as follows: $\begin{matrix}{\gamma_{s}:={\frac{r^{\overset{\_}{s}}}{r_{0}^{\overset{\_}{s}}} \leq {\gamma_{opt} + {\frac{{F^{\overset{\_}{s}}F^{s\quad\dagger}{\hat{r}}^{s}}}{r_{0}^{\overset{\_}{s}}}.}}}} & (13)\end{matrix}$Let β(s) denote the optimal residual attenuation factor, i.e.${\beta(s)} = {\frac{{\hat{r}}^{s}}{r_{0}^{\overset{\_}{s}}} \leq \gamma_{opt}}$and noting that this is independent of r₀ ^({overscore (s)}) (in factγ_(opt)=β({overscore (s)}), one can rewrite (13) asγ_(s)≦γ_(opt)+σ₁ [F ^({overscore (s)}) F ^(s†)]β(s)Based on Eq. (13), the location of the four optimal samples (s*) can bedetermined as follows (here, time compensation is ignored)$\begin{matrix}{s^{*} \in {\arg\quad{\min\limits_{{s} = 4}{\left\{ {\beta(s)} \right\}.}}}} & (14)\end{matrix}$

The four optimal samples (s*) can be obtained by looking at a plot ofthe absolute value of the optimal response ({circumflex over(r)}^({overscore (s)})), such as shown in FIG. 10, and picking the slotscorresponding to the four smallest values.

FIG. 10 is a graph illustrating suitable sample selection for monitoringtemperatures in a multi-zone furnace in accordance with an aspect of thepresent invention. An x-axis represents slot positions from 0 to 120 anda y-axis depicts optimal absolute error profile in Angstroms per minute.In FIG. 10, there are four relative minimum selection pointscorresponding to slot 13 (1001), slot 37 (1002), slot 85 (1003), andslot 102 (1004). This is in contrast to the slots corresponding to themaxima of the temperature responses (denoted as s_(max)) in FIG. 8,which correspond to slots 9, 27,102, and 110. FIG. 11 is a graphillustrating a comparison between error from optimal sampling (s*) anderror induced by picking the slots corresponding s_(max) in accordancewith an aspect of the present invention. An x-axis represents slotposition from 0 to 120 and a y-axis depicts corrected deposition rateprofile in Angstroms per minute. The attenuation factors for these casesare: (note that γ_(opt)=0.1863) γ_(s*)=0.1864 (optimum), and γ_(s)_(max) =0.2159 and result in corresponding curves 1101 based on optimumsampling and 1102 based on s_(max) sampling. It is appreciated by theinventor of the present invention that even though r₀ ^({overscore (s)})is zero-mean, the corrected rate profile in non-zero mean withμ(r^(s))=μ(F^(s)ΔT^(s)). This induces coupling between the temperatureand time loops. In fact, even though the profile derived from s_(max)exhibits less peak to peak variation than the optimal profile, it showsa significant shift in the mean rate. This will result in large changesin deposition time, which will repeat from run-to-run (i.e., eachiteration). To smoothen out the rate profile using s_(max) samples (theline 1102) will cause a large shift in deposition rate, which isundesirable for deposition time control.

Note that an alternate bound for γ_(s) can be derived as follows:$\gamma_{s} \leq {\gamma_{opt} + \frac{{F^{\overset{\_}{s}}F^{s\quad\dagger}{\hat{r}}^{s}}}{r_{0}^{\overset{\_}{s}}}} \leq {\gamma_{opt} + {{{F^{\overset{\_}{s}}F^{s\quad\dagger}}}\gamma_{opt}}} \leq {\left( {1 + {{\sigma_{1}\left( F^{\overset{\_}{s}} \right)}{\sigma_{1}\left( F^{s\quad\dagger} \right)}}} \right)\gamma_{opt}} \leq {\left( {1 + \frac{\sigma_{1}\left( F^{\overset{\_}{s}} \right)}{\sigma_{4}\left( F^{s} \right)}} \right){\gamma_{opt}.}}$

Temperature Control and Stability to Modeling Errors

The following discussion presents a closed-loop control scheme that canbe employed by a controller of the present invention. The followingdiscussion also analyzes bounds on allowable modeling errors. The latteris important as in practice it is extremely time consuming and costly torun characterization experiments on all the furnaces. Furthermore, wafersubstrate differences between the monitors used for deriving thetemperature response model, and product wafers at isolation and sidewallnitride deposition will cause the temperature response models to deviatefrom those derived above. Such deviation can be due to differences innucleation rates, as well as, changes in wafer surface area, and thermalmass.

First consider the feedback control scheme. At run k, let s_(k) denotethe sampling plan (in order to control temperature only, one needs|s_(k)|≧4, and for time and temperature, |s_(k)|≧5), T_(k) the 4temperature setpoints, r_(k) ^(s) ^(k) the zero-mean rate profile. Thenthe rate profile model can be expressed asr _(k) ^(s) ^(k) =α_(k) ^(s) ^(k) +F ^(s) ^(k) (T _(k) −T _(c))+η_(k)^(s) ^(k)where α_(k) ^(s) ^(k) ε

^(s) ^(k) ^(|) is the disturbance input, and η_(k) ^(s) ^(k) ε

^(s) ^(k) ^(|) is the sampled disturbance (generated from an i.i.d.distributed random variable η_(k)ε

^({overscore (s)}|). S_(k) can be treated as a random variable, and inorder to study stability, it is assumed that the sampling plan is fixed,i.e. s_(k)=s=μ(s_(k)) . The impact of this assumption will be assumed toresult in an additional component to the measurement noise, and giventhe independence of the variables to the sampling plan, the rate profilemodel can be written asr _(k) ^(s)=α_(k) ^(s) +F ^(s)(T _(k) −T _(c))+{tilde over (η)}_(k)^(s)  (15)where {tilde over (η)}_(k) ^(s) now comprehends the additional noiseinduced by variations in the sampling plan. The controller can now bederived as follows. Let {circumflex over (α)}_(k) ^(s) denote theestimate of disturbance input α_(k) ^(s) at run k. Then given theobservations ({T_(k),r_(k) ^(s)}) at run k, the estimate of α_(k) ^(s)at run k+1 can be obtained via an exponentially weighted moving average(EWMA) algorithm, where given 0<λ<2, one has{circumflex over (α)}_(k+1) ^(s)=(1−λ){circumflex over (α)}_(k) ^(s)+λ(r_(k) ^(s) −F ^(s)(T _(k) −T _(C))).  (16)Furthermore, given {circumflex over (α)}_(k) ^(s) at run k, thetemperature setpoint settings are obtained asT _(k) =−F ^(s†){circumflex over (α)}_(k) ^(s) +T _(c).  (17)The above equations can be combined to yield an integral controller fordetermining setpoint temperature adjustments as follows: $\begin{matrix}{{{{- F^{s\quad\dagger}}{\hat{\alpha}}_{k + 1}^{s}} = {{{- \left( {1 - \lambda} \right)}F^{s\quad\dagger}{\hat{\alpha}}_{k}^{s}} - {\lambda\quad F^{s\quad\dagger}r_{k}^{s}} + {\lambda\left( {T_{k} - T_{C}} \right)}}}{{which}\quad{using}\quad(17)\quad{yields}}\begin{matrix}{{T_{k + 1} - T_{C}} = {{\left( {1 - \lambda} \right)\left( {T_{k} - T_{C}} \right)} - {\lambda\quad F^{s\quad\dagger}r_{k}^{s}} + {\lambda\left( {T_{k} - T_{C}} \right)}}} \\{= {T_{k} - T_{C} - {\lambda\quad F^{s\quad\dagger}r_{k}^{s}}}}\end{matrix}{{which}\quad{finally}\quad{gives}}{T_{k + 1} = {T_{k} - {\lambda\quad F^{s\quad\dagger}r_{k}^{s}}}}} & (18)\end{matrix}$In practice, large values of λ are not desirable due to the possibilityof instability due to modeling errors. To characterize the impact ofthese errors, assume that the true furnace response is given by{overscore (F)}^(s) ⁵, and assume this can be expressed as {overscore(F)}^(s) =F ^(s) +E ^(s), where E^(s) denotes the modeling error.

FIG. 12 is a diagram illustrating a closed loop control scheme 1200 fordetermining setpoint adjustments in accordance with an aspect of thepresent invention. The term {circumflex over (T)}=T−T_(C) illustratesdeviation from a central temperature zone T_(C). A controller 1201inputs values in the form normalized deposition rate r^(s), disturbanceinput α^(s), and sampled disturbance {overscore (η)}^(s) and, basedthereon, generates temperature setpoint adjustments {overscore (T)},which are provided the a multi-zone furnace system 1202 that altersetpoint temperatures of one or more zones of the multi-zone furnacesystem 1202 while maintaining a setpoint temperature of the centraltemperature zone at a nominal value. The input values are computed fromtemperature sample measurements s_(k).

FIG. 12 illustrates the feedback loop described by Eq. 18 and depictsmeasured rate deviations from the mean rate that are compared to 0 andfed back into the controller. From this, the controller 1201 sends anupdated temperature setpoint vector {circumflex over (T)} to thefurnace. The setpoint temperatures of the one or more zones aretypically not at T_(C). Additionally, observations/measurements aresubject to measurement noise, represented by the term {overscore(η)}^(s), the sampled disturbance. As a result, the inherent rateprofile α^(s) (which is the rate profile in the furnace when all zonesare at T_(C) (as shown, for example in FIG. 6)) is not known and istreated as a disturbance input.

From FIG. 12, in order to ensure stability, the following is true:$\begin{matrix}{{{{\det\left( {I_{s} + \frac{\lambda\quad{\overset{\_}{F}}^{s}F^{s\quad\dagger}}{z - 1}} \right)} \neq 0},{{{for}\quad{all}\quad{z}} > 1.}}{{This}\quad{implies}\quad{that}}\quad{{\det\left( {{zI}_{s} - \left( {I_{s} - {\lambda\quad{\overset{\_}{F}}^{s}F^{s\quad\dagger}}} \right)} \right)} \neq {0\quad{for}\quad{all}\quad{z}} > 1.}{{Which}\quad{in}\quad{turn}\quad{holds}\quad{if}}{{\rho\left( {I_{s} - {\lambda\quad{\overset{\_}{F}}^{s}F^{s\quad\dagger}}} \right)} \leq 1}{or}{0 \leq \left\{ {{eig}\left( {{\lambda\left( {F^{s} + E^{s}} \right)}F^{s\quad\dagger}} \right)} \right\} \leq 2}{or}{{{\lambda\sigma}_{1}\left( {E^{s}F^{s\quad\dagger}} \right)} \leq {\min\left\{ {\lambda,{2 - \lambda}} \right\}}}{or}{{{\sigma_{1}\left( E^{s} \right)} \leq {\min\left\{ {1,\frac{2 - \lambda}{\lambda}} \right\}\frac{1}{\sigma_{1}\left( F^{s\quad\dagger} \right)}}} = {\min\left\{ {1,\frac{2 - \lambda}{\lambda}} \right\}{{\sigma_{4}\left( F^{s} \right)}.}}}} & (19)\end{matrix}$Eq. (19) also shows the importance of picking a sampling plan thatyields a sufficiently high σ₄[F^(s)]. Compare this to the abovediscussion, which indicates that in order to keep the error induced dueto sampling small, a sampling plan should be picked with a largeσ₄[F^(s)]. Both of these conditions indicate that it is important topick a sampling plan that yields a bounded σ₄[F^(s)] i.e.σ₄(F^(s))>κ_(min), where κ_(min)>0 is a predefined limit. This ensuresadaptability to and mitigation of modeling errors, as well as, minimizesthe error induced by sampling. It is also noted that stability ismaintained as 1 is an eigenvalue of (I_(|s|)−λ{overscore (F)}^(s)F^(s†))for any s (measurement samples) such that |s|>4.

Iterative Implementation

The following discussion combines time adjustments and temperatureadjustments into an iterative controller scheme that mitigates undesireddeposition rate variations. The scheme is presented to track and updateoff batch data. The scheme is iterative in order to monitor and trackproduct lots in any batch within the controller. For illustrativepurposes, some details are omitted in order to keep the presentationclear, such as, for example, those dealing with batch tracking,mis-sequenced batches, measurement delays, and numerical tests on matrixconditioning, amongst others. As noted above, a non-optimal samplingplan for optimizing temperatures may be employed instead of an optimalsampling plan. For example, there are five setpoints (four temperatures,and deposition time) to track in the present example, five measurementsfrom five slots of the furnace are required. This, however, causes thesampling plan to deviate from the minimal optimal plan s*. The fifthslot should ideally be selected to be the least sensitive one totemperature changes. This is obtained by finding the slot correspondingto the minimal norm row of F^({overscore (s)}). For the furnace responseconsidered so far this corresponds to slot 67. Once again in practicethis exact selection may not be possible.

As noted in the previous discussion, the mean of the deposition rateafter temperature adjustment is in general non-zero and is given byμ[F^(s)ΔT^(s)]. This rate correction needs to be applied to the Kalmanfilter Eq. (5). Let N be the total number of expected measurements froma batch (e.g., if a single wafer per lot is measured then N would benumber of lots). Given these N measurements, the rate profile isapproximated as being zero-mean by transforming the measured rates todeviation from their mean. Product wafers are measured on a lot-by-lotbasis (vs. being batched). This forces two levels of updates, one fromlot-to-lot (i); and the other from batch-to-batch (k). Let s_(i) denotethe slot number for measurement i. Also, let${f\left( s_{i} \right)} = {\begin{bmatrix}{f^{1}\left( s_{i} \right)} \\{f^{2}\left( s_{i} \right)} \\{f^{3}\left( s_{i} \right)} \\{f^{4}\left( s_{i} \right)}\end{bmatrix}.}$

Now consider the following feedback algorithm, which can be employed bymethods and/or systems of the present invention to control depositiontime and temperature: let i:=0; FI(0) := 0_(4×4); FY := 0_(4×1); Ξ_(F)(0) := 0_(4×1); {overscore (r)}(0) := 0. while (i < N) do FI(i + 1) :=FI(i) + f(s_(i))f′(s_(i)). FY(i + 1) := FY(i) +y_(k)(s_(i))f(s_(i))/t_(k). Ξ_(F)(i + 1) := Ξ_(F)(i) + f(s_(i)).${\overset{\_}{r}\left( {i + 1} \right)}:={\frac{{i{\overset{\_}{r}(i)}} + {{y_{k}\left( s_{i} \right)}/t_{k}}}{i + 1}.}$i := i + 1. if (i = N) T_(k+1) := T_(k); ΔT_(k) := 0. if (N ≧ 5) Computeσ₄ (FI(N)). //error metric if (σ₄(FI(N)) > κ_(min) ²) // if metricgreater than deadband T_(k+1) := T_(k) − λFI⁻¹(N)(FY(N) −Ξ_(F)(N){overscore (r)}(N)). // update temp setpoints ΔT_(k) :=−λFI⁻¹(N)(FY(N) − Ξ_(F)(N){overscore (r)}(N)). endif endif Update Σ, K,and ξ via Eq. (5). Update {circumflex over (θ)} via (Eq. 5 updatesKalman filter and excitation) {circumflex over (θ)}_(k+1) := {circumflexover (θ)}_(k) + K_(k)(({overscore (r)}(N) + Ξ_(F) ^(′)(N)ΔT_(k)/N)t_(k)− u_(k)θ_(k)). // update deposition times endif endwhile

Note that in the iterative scheme above, the update for {circumflex over(θ)} has been modified to include a correction term which representsμ[F^(s)ΔT^(s)]. Furthermore, the temperature update involvestransforming the observed rates to offsets from their mean value. Inorder to simplify the computation, the singular value check on F^(s) hasbeen replaced by a check on the singular value of (F^(s′)F^(s)), wherethe latter is the square of the former (and hence the test is doneagainst κ_(min) ², which is used to ensure that the model is sufficientto overcome mismatch in actual furnace behavior, as well as, mitigatingthe impact of furnace subsampling on the cross-load error.

The update scheme presented in this section sidelines issues related tooutlier detection, measurement delays, mis-sequenced measurements,maintenance events (e.g., furnace liner changes), updates onqualification runs, and monitoring of controllers to preventmis-process. This has been done to keep the overall presentation asclear as possible. Modifications need to be made to the update scheme toaddress these issues prior to implementation in production.

PRODUCTION EXAMPLE

The following discussion describes a production example in which timeand temperature control is applied to multiple furnaces running sidewallnitride and isolation nitride deposition in high-volume production. Forpurposes of control, the same temperature response model (as derived forFIG. 8) is used across all furnaces for both the isolation and sidewallnitride processes. The furnaces are vertical furnaces with fivetemperature zones and are capable of running four production lots at atime, and the thickness measurements are obtained via interferometery.Data is transferred to and from the tools to a factory host where allcontrol computations are performed. Updated time and temperaturesetpoints are downloaded at the start of a batch. Each furnace iscapable of holding four 24 wafer product lots per batch. A single waferfrom each lot is measured with respect to deposition rate afterdeposition. In order to generate 5 samples per batch, the monitor in thehighest slot is always measured. Due to deposition rate differencesbetween monitor and product wafers, a correlation offset is used toconvert the monitor thickness into an equivalent product thickness.

FIG. 13 is a histogram illustrating distribution of probability ofsampling specific slots across the furnace per lot in accordance with anaspect of the present invention. An x-axis represents furnace slotposition ranging from 0 to 120 and a y-axis represents probability ofsampling. The histogram is obtained by observing location of the productwafers measured while the furnace is running production. The sampleswith the greatest probability correspond to slots 12, 36, 61, 85, and108, where the last slot is that of the monitor wafer. The attenuationgain corresponding to these slots is calculated (with respect to a zeromean linear profile) as γ_(s)=0.1935 which places it close to theoptimal value (γ_(opt)=0.1863) obtained when all product wafers in theload are assumed to be sampled. Furthermore, for these slotsσ₄(F^(s))=0.066.

FIG. 14 is a histogram illustrating distribution of σ₄(F^(s)) (an errormetric) based on the sampling distribution shown in FIG. 13 inaccordance with an aspect of the present invention. An x-axis representsfurnace slot position ranging from 0 to 120 and a y-axis represents aprobability of σ₄(F^(s)). A suitable threshold or dead zone value isobtained at κ_(min)=0.05 at 1401. For any λ≦1 in (16), this givesσ₁(E^(s))<0.05 via (19) which corresponds to a 15% allowable variationin the model (as measured by σ₁(E^(s))/σ₁(F^(s)) for the most probableslot sampling from FIG. 14). Since the production slots measured are notpredetermined, the threshold test on σ₄(F^(s)) prevents the models fromupdating if σ₄(F^(s)) is too small for that specific batch. It is ofinterest to note that such a choice also excludes the sampling plans_(max)(see FIG. 11), which results in large shifts in mean depositionrate. This is done to ensure there is always sufficient margin (19) toguarantee stable convergence of temperatures, while also making sure(via (13)) that the error induced due to sampling is limited. Byensuring sufficient margin, the same temperature response model can beemployed across processes involving different substrate conditions(which impact deposition rate), and also across multiple process tools.

FIGS. 15 and 16 show the result of applying the control scheme atisolation and sidewall nitride processes. FIG. 15 is a graphillustrating deviation from expected target thicknesses for depositedisolation nitride in accordance with an aspect of the present invention.An x-axis depicts numbers of lots and a y-axis depicts percent errorfrom target deposition thicknesses. Line 1501 illustrates processperformance with iterative temperature and time control of the presentinvention and line 1502 is without the iterative temperature and timecontrol. It can be seen that the line 1501 shows substantially lessdeviation from target thicknesses than does the line 1502. FIG. 16 is agraph illustrating deviation from expected targets for depositedsidewall nitride in accordance with an aspect of the present invention.Again, an x-axis depicts numbers of lots and a y-axis depicts percenterror from target deposition thicknesses. Line 1602 illustrates processperformance with iterative temperature and time control of the presentinvention and line 1601 is without the iterative temperature and timecontrol. It can be seen that the line 1602 shows substantially lessdeviation from target thicknesses than does the line 1601. The sidewallnitride shows greater variability as a percent of target, not justbecause the deposited film is thinner, but also due to device patterninduced metrology noise. Performance improvements as measured by theroot-mean-square (rms) of the percent error are as follows: The rmspercent error for isolation nitride drops from 0.86% to 0.18%. The rmspercent error for sidewall nitride drops from 1.15% to 0.44%.

In addition, additional furnace capacity can be obtained due to a dropin furnace idle time while an excursion is detected, and manualadjustments made. FIG. 17 and 18 shows a time and temperatureadjustments (all arbitrarily shifted) for isolation nitride on a singlefurnace. FIG. 17 is a graph illustrating deposition time adjustments fora number of batches in accordance with an aspect of the presentinvention. An x-axis represents numbers of batches and a y-axisrepresents adjustments to deposition time using the control mechanism ofthe present invention. FIG. 18 is a graph illustrating temperaturesetpoint adjustments for a number of batches in accordance with anaspect of the present invention. An x-axis represents numbers of batchesand a y-axis represents adjustments to deposition temperature setpointsusing the control mechanism of the present invention. It can be seenthat the time compensating for a rate drift across multiple furnaceruns, and includes a shift on a furnace liner change, while thetemperature changes correct for drifts in specific furnace zones.

The above discussions describe a scheme for control of deposition timeand temperature for LPCVD silicon nitride. An analysis of furnacecontrollability, and the impact of slot selection for measurement(sampling) on process performance were characterized. Stability of thefeedback loop to modeling errors while constructing the temperatureresponse model has been quantified. The quantified results are employedto enable the same model to be used across multiple tools, and acrossdifferent substrates (which impacts the deposition rate). Furthermore,this also bounds the error induced due to sampling. An iterativealgorithm was presented for tracking batch data and to update frombatch-to-batch.

Although the invention has been shown and described with respect to acertain aspect or various aspects, it is obvious that equivalentalterations and modifications will occur to others skilled in the artupon the reading and understanding of this specification and the annexeddrawings. In particular regard to the various functions performed by theabove described components (assemblies, devices, circuits, etc.), theterms (including a reference to a “means”) used to describe suchcomponents are intended to correspond, unless otherwise indicated, toany component which performs the specified function of the describedcomponent (i.e., that is functionally equivalent), even though notstructurally equivalent to the disclosed structure which performs thefunction in the herein illustrated exemplary embodiments of theinvention. In addition, while a particular feature of the invention mayhave been disclosed with respect to only one of several aspects of theinvention, such feature may be combined with one or more other featuresof the other aspects as may be desired and advantageous for any given orparticular application. Furthermore, to the extent that the term“includes” is used in either the detailed description or the claims,such term is intended to be inclusive in a manner similar to the term“comprising.”

1. A system comprising: a multi-zone furnace having a plurality oftemperature zones and a central temperature zone; and a controller thatreceives a number of thickness measurements from the multi-zone furnaceas feedback and iteratively adjusts zone temperature setpoints for theplurality of temperature zones and deposition times of the multizonefurnace according to a thermal model and the received measurements. 2.The system of claim 1, wherein the thermal model includes depositionrates for the plurality of zones according to deviations in temperaturefrom the central temperature zone.
 3. The system of claim 1, wherein thecontroller additionally controls pressure of the multi-zone furnace andgas flow into the furnace of a deposition material source.
 4. The systemof claim 1, further comprising a metrology tool that provides depositionprocess parameters to the controller, wherein the controller initiallysets the zone temperature setpoints for the plurality of temperaturezones and the central temperature zone at least in part according to theprovided deposition process parameters.
 5. The system of claim 4,wherein the deposition process parameters include one or more targetthicknesses.
 6. The system of claim 1, wherein the multi-zone furnaceincludes a number of slots throughout the plurality of temperature zonesand the central temperature zone.
 7. The system of claim 6, wherein thenumber of slots are filled with monitor wafers, product wafers, anddummy wafers.
 8. The system of claim 7, further comprising probes thatobtain the thickness measurements from selected product wafers andprovide the thickness measurements to the controller.
 9. The system ofclaim 1, wherein the controller determines an error metric according tothe thermal model and the received measurements that indicate modelerror.
 10. The system of claim 1, wherein the controller determines anerror metric according to sub-sampling error and adjusts the zonetemperature setpoints on the error metric being above a dead zone value.11. The system of claim 1, wherein the controller builds the thermalmodel of deposition rates for the plurality of zones from thicknessmeasurements.
 12. The system of claim 1, wherein the controllerdetermines an excitation value according to the thickness measurementsthat indicates presence of information available in the thicknessmeasurements.
 13. The system of claim 12, wherein the controllerdetermines the excitation value by tracking changes processes performedby the furnace.
 14. The system of claim 12, wherein the controlleradjusts the deposition times at least in part according to theexcitation value.
 15. A furnace controller comprising: a thermal modelthat provides changes in deposition rates according to temperaturechanges in a number temperature zones; a filter that recursivelydetermines adjustments to deposition time according to deposition ratesin the multiple temperature zones; an update component that receives anumber of thickness measurements, employs the thermal model to determinesetpoint temperature values for the number of temperature zones,determines deposition rates from the number of thickness measurements,and employs the filter to obtains updates to the deposition time. 16.The furnace controller of claim 15, wherein the number of measurementsis at least equal to the number of temperature zones.
 17. The furnacecontroller of claim 15, wherein the update component identifieserroneous thickness measurements.
 18. The furnace controller of claim15, wherein the update component adjusts the number of thicknessmeasurements according to variations in a present furnace load and aprevious furnace load on which the thermal model is based.
 19. Thefurnace controller of claim 15, wherein the filter is a Kalman filter.20. A method of controlling a furnace during deposition comprising:obtaining a number of thickness measurements; determining an errormetric according to the number of thickness measurements that indicatessubsampling error; on the error metric being acceptable, using a thermalmodel to determine adjusted setpoint temperatures for a number of zonesof the furnace according to the thermal model, the number of thicknessmeasurements, and one or more target thicknesses; determining anexcitation value according to batch to batch variations in furnaceoperation; adjusting one or more parameters of a filter based on thedetermined excitation value; and adjusting a deposition time accordingto the filter.
 21. The method of claim 20, wherein building the thermalmodel comprises determining adjustments in deposition rate per slot ofthe furnace according to temperature changes in the number of zones ofthe furnace.
 22. The method of claim 19, wherein the one or more targetthicknesses include an isolation layer target thickness and a sidewalltarget thickness.
 23. The method of claim 20, further comprisingbuilding a thermal model that estimates deposition rate change accordingto temperature change.
 24. The method of claim 20, further comprisingselecting a number production wafers located in associated slots toprovide the thickness measurements.
 25. The method of claim 24, whereinthe error metric is further determined according to locations of theassociated slots that provide the thickness measurements.
 26. The methodof claim 20, wherein a subset of the one or more parameters are updatedon the excitation value being above a threshold value and remainingparameters of the one or more parameters are maintained at previousvalues.
 27. The method of claim 20, wherein the excitation value isdetermined according to changes in batch to batch thickness targets.